Found problems: 253
Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
I. Voronovich
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
Find the equation of the line tangent to the parabola $y = 1/3(x^2-2x+4)$ and a circle of unit radius centered at the origin. (List all solutions.)
Source: 1976 Euclid Part A Problem 4
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The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$
A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.
On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.
A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$?
$ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad
\textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad
\textbf{(C)}\ -\frac{4}{5} \qquad
\textbf{(D)}\ -\frac{3}{5} \qquad
\textbf{(E)}\ -\frac{1}{2} $
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
Line passes the focal point $F$ of parabola $y^2=8(x+2)$ with bank angle of $60^{\circ}$ intersects the parabola at $A,B$. Perpendicular bisector of $AB$ intersects $x$-axis at $P$, then the length of $PF$ is
$\text{(A)}\frac{16}{3}\qquad\text{(B)}\frac{8}{3}\qquad\text{(C)}\frac{16}{3}\sqrt3\qquad\text{(D)}8\sqrt3$
Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis.
You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.
We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is:
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 35\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 30\qquad
\textbf{(E)}\ \text{not finite}$
Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that
\[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0
\]The area of $ R$ is closest to
$ \textbf{(A)}\ 21 \qquad
\textbf{(B)}\ 22 \qquad
\textbf{(C)}\ 23 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 25$
Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum?
$ \textbf{(A)}\ \frac{a+b}{2} \qquad\textbf{(B)}\ a+b \qquad\textbf{(C)}\ \sqrt{ab} \qquad\textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad\textbf{(E)}\ \frac{a+b}{2ab} $
For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.
Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]
Let $ABCD$ be a square such that $AB$ lies along the line $y=x+8,$ and $C$ and $D$ lie on the parabola $y=x^2.$ Find all possible values of sidelength of the square.
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$
The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil
attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{infinitely many}$